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Paper Reference(s) 6668/01 Edexcel GCE Further Pure Mathematics FP2 Advanced Thursday 24 June 2010 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae (Pink) Items included with question papers Nil Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulas stored in them. Instructions to Candidates Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6668), your surname, initials and signature. Information for Candidates A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. Full marks may be obtained for answers to ALL questions. There are 8 questions in this question paper. The total mark for this paper is 75. Advice to Candidates You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit. N35388A This publication may only be reproduced in accordance with Edexcel Limited copyright policy. ©2010 Edexcel Limited. 1. (a) Express 3 in partial fractions. (3r 1)(3r 2) (2) (b) Using your answer to part (a) and the method of differences, show that n 3 (3r 1)(3r 2) r 1 = 3n . 2(3n 2) (3) 1000 (c) Evaluate r 100 3 , giving your answer to 3 significant figures. (3r 1)(3r 2) (2) 2. The displacement x metres of a particle at time t seconds is given by the differential equation d2x + x + cos x = 0. dt 2 When t = 0, x = 0 and dx 1 = . 2 dt Find a Taylor series solution for x in ascending powers of t, up to and including the term in t3. (5) 3. (a) Find the set of values of x for which x+4> 2 . x3 (6) (b) Deduce, or otherwise find, the values of x for which x+4> 2 . x3 (1) N35388A 2 z = −8 + (8√3)i 4. (a) Find the modulus of z and the argument of z. (3) Using de Moivre’s theorem, (b) find z3, (2) (c) find the values of w such that w4 = z, giving your answers in the form a + ib, where a, b ℝ. (5) 5. Figure 1 Figure 1 shows the curves given by the polar equations r = 2, and r = 1.5 + sin 3θ, 0θ , 2 0θ . 2 (a) Find the coordinates of the points where the curves intersect. (3) The region S, between the curves, for which r > 2 and for which r < (1.5 + sin 3θ), is shown shaded in Figure 1. (b) Find, by integration, the area of the shaded region S, giving your answer in the form aπ + b√3, where a and b are simplified fractions. (7) N35388A 3 6. A complex number z is represented by the point P in the Argand diagram. (a) Given that z − 6 = z, sketch the locus of P. (2) (b) Find the complex numbers z which satisfy both z − 6 = z and z − 3− 4i = 5. (3) The transformation T from the z-plane to the w-plane is given by w 30 . z (c) Show that T maps z − 6 = z onto a circle in the w-plane and give the cartesian equation of this circle. (5) 7. 1 2 (a) Show that the transformation z = y transforms the differential equation 1 dy – 4y tan x = 2 y 2 dx (I) dz – 2z tan x = 1 dx (II) into the differential equation (5) (b) Solve the differential equation (II) to find z as a function of x. (6) (c) Hence obtain the general solution of the differential equation (I). (1) N35388A 4 8. (a) Find the value of λ for which y = λx sin 5x is a particular integral of the differential equation d2 y + 25y = 3 cos 5x. dx 2 (4) (b) Using your answer to part (a), find the general solution of the differential equation d2 y + 25y = 3 cos 5x. dx 2 (3) Given that at x = 0, y = 0 and dy = 5, dx (c) find the particular solution of this differential equation, giving your solution in the form y = f(x). (5) (d) Sketch the curve with equation y = f(x) for 0 x π. (2) TOTAL FOR PAPER: 75 MARKS END N35388A 5